Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2856 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,12,16)(11,19,21,24,20,15,22,23)(25,26)(27,33,34,40)(28,30)(29,32)(31,35,38,37)(36,39);(1,3,4,8)(2,5,6,7)(11,20)(15,19)(21,22)(23,24)(26,30)(27,34)(32,39)(35,37);(3,8)(5,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,19)(18,24)(25,27)(26,31)(28,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,6)(5,7)(9,12,14,13)(10,16,17,18)(11,21,20,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122386);